Manual Homological Dimensions of Modules

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Join our email list. Sign up. Ordering on the AMS Bookstore is limited to individuals for personal use only. Homological Dimensions of Modules Share this page. Advanced search. Author s Product display : B. Abstract: These notes were prepared for a series of ten lectures given at Regional Conference of the Conference Board of the Mathematical Sciences in June Volume: Publication Month and Year: Copyright Year: Page Count: Cover Type: Softcover. Print ISBN Online ISBN Increases and decreases in leaf chlorophyll and N content were similar for both cultivars.

Likewise, multidimensional scaling MDS analysis indicated both cultivar-independent and cultivar-specific gene expression. However, many genes encoding nucleotide-binding leucine rich repeat NB-LRRs proteins and wall-bound kinases associated with detecting and responding to environmental signals were differentially expressed. Several of these belonged to unique cultivar-specific gene co-expression networks. It is plausible that cultivar ecotype -specific genes and gene-networks could be one of the drivers for the documented differences in responses to leaf-borne pathogens between these two cultivars.

Incorporating broad resistance to plant pathogens in elite switchgrass germplasm could improve sustainability of biomass production under low-input conditions. Switchgrass Panicum virgatum L. Two ecotypes of switchgrass are known, with the lowland ecotypes adapted to wetter regions, and the upland ecotypes adapted to drier conditions. Upland and lowland cultivars of switchgrass differ in their genetics [ 2 — 7 ], yield potential [ 8 — 13 ], upland cultivars are better adapted for cold, winter survival [ 14 — 16 ], and lowland ecotypes exhibit higher resistance to certain pathogens [ 17 — 19 ].

More specifically, as a source of biomass, switchgrass is targeted to be cultivated with minimal external inputs on marginal soils that are not best suited for row crops [ 8 , 20 , 21 ]. Sustainable production of biomass from switchgrass is dependent on stable yields over several years and good local adaptation of the cultivars within their regions of production.

Because infection by viral pathogens can drastically suppress growth in susceptible switchgrasses [ 19 ] and significant losses in ethanol yields can result from severe infestations by rust fungi Puccinia emaculata [ 22 ], there is uncertainty of yield stability of switchgrass under disease pressures. Plants contain several mechanisms for monitoring the environment, with several specifically evolved for detecting pathogens [ 23 ].

Genes encoding nucleotide-binding leucine rich repeat proteins NB-LRR comprise the largest numbers of disease resistance genes R genes known in plants [ 24 — 27 ]. R genes are usually part of large gene families with many expressed at low levels in plant cells, suggestive of their role in monitoring [ 28 ]. Because NB-LRRs can participate in protein-protein interactions and are present in large complexes, it is likely they recognize changes in host proteins that respond to pathogen elicitors and subsequently catalyze downstream reactions [ 28 — 30 ].

Once a NB-LRR-catalyzed signaling cascade is triggered, it results in a significant redirection of plant metabolism with a range of resistance responses [ 31 ]. Although several of these resistance-related processes have been investigated in other plants, there is considerable lack of data in switchgrass and related perennial feedstocks. Frazier et al. Among the findings consistent with data presented by Uppalapati et al. Several other classes of genes and the proteins they encode, such as wall-bound kinases, NADPH-oxidases, protein kinases, protein phosphatases, and transcription factors also participate in monitoring environmental changes in plants either directly or indirectly via interactions with primary signals [ 33 — 35 ].

Among transcription factors, genes encoding WRKYs respond strongly to environmental stress, and play a significant role in plant immunity [ 36 ]. Switchgrass contains at least WRKY genes [ 37 ]. Several of these WRKYs were associated with specific developmental stages of flag leaves in field-grown switchgrass, and 23 WRKY genes were associated with a senescence-associated gene co-expression module [ 37 ].

An RNA-Seq study of flag leaf development in field-grown switchgrass developed a framework to understand the molecular signatures associated with development through senescence onset for switchgrass [ 38 ]. Other similar studies have resulted in the identification of specific NAC transcription factors that can positively modulate leaf senescence in switchgrass [ 39 , 40 ].

Upland and lowland leaf transcriptomes have also been analyzed by RNA-Seq [ 41 ], although these authors only analyzed a single leaf from one individual genotype of three cultivars at one harvest date, making developmental and upland versus lowland comparisons difficult. However, their data indicated that transcripts associated with photosynthesis and cellular components related to photosynthesis were significantly more abundant in the lowland leaves relative to the single upland leaf analyzed.

Additionally, several genes potentially involved with plant defense, such as those encoding catalase, S-adenosylmethionine synthase, and wound-induced protein were differentially regulated in the upland leaf sample as compared with the lowland leaf samples.

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How representative these changes were across genotypes or development were not explored. Stabilized half-sib families arising from such hybrids outperformed the maternal Summer population for yield and outperformed the paternal Kanlow population for winter survival. However, several Summer x Kanlow hybrid switchgrass plants, including the cultivar Liberty, suffered from increased disease pressure as compared to the Kanlow parent [ 19 ], suggesting that potential disease-related traits in Kanlow were not yet incorporated into hybrid progeny. Moreover, there are limited data on the genes related to surveillance mechanisms and pathogen defense in Kanlow.

This study was undertaken to determine the temporal changes in leaf gene expression in Kanlow and Summer plants grown under controlled greenhouse conditions. The goals were to: 1 develop a foundational dataset of the gene co-expression networks that impact overall leaf functions; 2 discover similarities and differences in NB-LRR and receptor-like kinase RLK genes associated with surveillance and defense in Kanlow and Summer plants; 3 distinguish co-expression modules that were shared or unique to either cultivar; 4 identify some NB-LRR genes with population-specific transcription.

Seeds of the two tetraploid switchgrass cultivars, Kanlow and Summer, were obtained from plants grown in field nurseries. Both of these cultivars are synthetic populations derived from a collection of lowland and upland plants [ 44 ]. Plants were raised from seeds planted in cone-tainers 3. Prior to planting, cone-tainers were well watered and lightly tapped to settle soil and remove air gaps. Several seeds were distributed on top of the wetted soil mixture, lightly covered with a layer of the dry soil mixture and gently watered. Two weeks following germination, plants were thinned to leave one seedling per cone-tainer.

At this time, cone-tainer racks were moved to large plastic tubs to facilitate water absorption through the soil. Approximately plants were tagged upon the emergence of the third leaf on the primary tiller to provide adequate plants at similar leaf developmental stages for the duration of sampling. Sampling was confined to the 4 th leaf on the primary tiller, with new plants used at each sampling date.

Plants were discarded once the 4 th leaf was excised. Leaves were collected following emergence until late visible senescence for a total of seven harvests that occurred over a 60 day period. The first sampling date D0 was at the emergence of the 4 th leaf, successive harvests occurred at approximately 7—13 day intervals. Leaves were collected from three replicate samples for a total of 42 samples 2 populations x 7 harvest dates x 3 replicates , each containing 15 randomly selected individual plants per replicate to maximize population-specific differences and minimize genotype-specific effects.

The 15 excised leaves per replicate were pooled, cut into approximately 6 to 8 cm pieces, placed within 50 mL polypropylene tubes, capped, and flash frozen with liquid N 2. All samples were cryogenically ground with liquid N 2 using mortars and pestles. Aliquots of approximately 50 mg of ground material were used for total chlorophyll measurements [ 38 ]. For total N analyses, approximately mg of ground leaf samples were transferred to borosilicate glass tubes 0.

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Aliquots of approximately 10 mg of oven-dried samples were analyzed for total N [ 45 ]. Forty-two individual libraries were pooled with a loading concentration of 1. Run quality was monitored using Basespace Illumina, Inc. Trimmed reads were then aligned to version 4. Samtools was used to convert alignments to sorted BAM files [ 49 ] and gene expression counts were calculated for reads uniquely mapped to exons using featureCounts [ 50 ].

Multi-dimensional scaling MDS plots were generated using the metaMDS function in the vegan package [ 51 ] in R [ 52 ] with Euclidean distance measures. Prior to differential expression analysis, lowly-expressed genes were removed from the dataset by requiring each gene to have more than two counts per million CPM in at least three of the 42 total samples. Read pairs mapped to gene coding regions were counted using featureCounts [ 50 ] and used for calculating genomic coverage for each gene in the Kanlow and Summer populations.

The software IGV was used for genomic coverage visualization [ 57 , 58 ].

Putative receptor-like kinases RLKs were identified in the switchgrass genome using hmmsearch version 3. Chlorophyll content and total N were used as reliable indicators of leaf development and senescence for both cultivars. Chlorophyll content increased between the first and third harvests in Summer leaves and decreased significantly thereafter Fig 1A.

Although peak chlorophyll content was higher in Summer leaves it was not significantly different than peak chlorophyll content in Kanlow leaves. A chlorophyll and B total N. The 4 th leaf was harvested from upland Summer and lowland Kanlow switchgrass cultivars starting at emergence D0 followed by six successive harvests until late senescence D Summer blue bars ; Kanlow orange bars. Changes in total leaf N content were similar for both cultivars, where peak N content was observed in emerging leaves, D0 harvests. Total leaf N decreased subsequently at each successive harvest with lowest total leaf N observed at the last D60 harvest Fig 1B.

Overall, there was an average of However, a significant difference in mapping rates was observed between the Kanlow and Summer populations with Kanlow having higher mapping Leaf transcriptomes were subjected to MDS analysis. Transcriptomes of Summer leaves were distinguished from those of Kanlow leaves in the first dimension Fig 2A , but the overall change in transcriptome profiles across leaf development was similar in the second dimension. Curiously, the Kanlow D0 transcriptomes red circles, Fig 2A were distinguished from all the other transcriptomes.

These similarities and differences in transcriptome profiles suggested both cultivar-specific and cultivar-independent gene expression as potential underlying causes. A MDS analysis and B gene expression heatmap. Harvest dates as described for Fig 1.

In panel A, Kanlow transcriptomes are in circles and Summer transcriptomes are in triangles. Each harvest date is color coded. In panel B, one way clustering was performed and the dendrograms removed for ease of presentation. To further explore these nuances in gene-expression profiles, heatmaps of differentially expressed genes DEGs were constructed to visualize global gene expression profiles Fig 2B.

Emerging Kanlow 4 th leaves D0 contained a cluster of highly expressed genes that were expressed in much lower levels across all the other time points in both cultivars. At all subsequent time points, individual harvest date comparisons of transcriptomes more likely reflected basal ecotype and time-dependent driven differences in expression. Analysis of DEGs with or without the Kanlow D0 datasets did not appreciably affect these overall findings, suggesting that the Kanlow D0 transcriptomes were part of normal leaf development.

Small discrepancies in leaf emergence and developmental stages between Summer and Kanlow plants at the D0 sampling point could have accounted for these differences. The gene expression profiles of metabolic processes central to leaf function were evaluated for the two contrasting cultivars by summing the transcript counts of paralogous genes with the same protein annotation for example: PEP carboxylase, glutamine synthetase, etc.

With minor variations outside of the D0 Kanlow transcriptomes all these processes followed a similar trend Fig 3. Genes encoding proteins required for chlorophyll biosynthesis were elevated at the first two harvests in Summer and the second harvest in Kanlow Fig 3A , and thereafter declined in parallel in both cultivars. Similarly, expression of genes associated with chlorophyll degradation increased over time, with a jump in expression recorded between the D37 and D49 harvest dates, probably coincident with the onset of leaf senescence Fig 3B.

Likewise, expression of genes required for photosystems Fig 3C , light harvesting complexes Fig 3D , and the Calvin cycle Fig 3E mirrored the trend seen for expression of genes associated with chlorophyll biosynthesis Fig 3A. Both differentially expressed and non-differentially expressed transcripts from genes with at least normalized counts at one time point were included.

Paralogous genes with low counts were excluded in these analyses. Counts for individual transcripts were converted to z-scores and averaged for each time point. Highest expression of these genes was seen at the last two harvest dates D49 and D60; Fig 3J , when leaves were visibly senesced and had lost significant amounts of chlorophyll Fig 1A. The identities of the genes associated with the pathways described in Fig 3 are given in S1 Data.

Additionally, there was a significant temporal upregulation of many switchgrass homologs of Arabidopsis senescence-associated genes SAGs during flag leaf senescence [ 38 ]. For heat maps, one way clustering was performed and the dendrograms removed for ease of presentation. Gene identities are provided in S1 Data. Although there were some minor differences in the expression of these NACs between the two cultivars, the overall patterns were similar, suggesting that the timing and progression of senescence were similar in Kanlow and Summer leaves.

A majority 52 out of 74 of the SAG genes had profiles consistent with association with leaf senescence as they were most highly expressed at the last two harvest dates D49 and D60; Fig 4B ; S1 Data.

On the existence of certain modules of finite Gorenstein homological dimensions, II

However, the other 22 switchgrass SAG homologs S1 Data did not display an expression pattern consistent with an association with leaf senescence. Whether, this arises because of an imperfect orthology assignment during mapping onto the switchgrass genome or due to cultivar and ecotype differences in gene expression remains unclear. Similar and dissimilar relative expression of several SAGs were documented between Kanlow and Summer leaves. As examples of genes with similar expression profiles in the two ecotypes were Pavir. Several other SAGs were expressed at a higher level in senescing Kanlow than in senescing Summer leaves: these included Pavir.

Genes that were more abundantly expressed in senescing Summer leaves than in Kanlow leaves included Pavir. These data highlight the preferential upregulation of specific loci in the two cultivars and provide some evidence for subtle differences in the execution of the leaf senescence program between Summer and Kanlow plants. Gene co-expression network modules referred to as M in the text determined by analyses of the RNA-Seq datasets are shown in Fig 5. All other data associated with these analyses are provided in S1 Data. Modules were broadly separated into 3 categories: 1 those in which the two cultivars shared a similar profile of gene expression over time M5, M6, M10, M13, M14, M15, and M16 or those differing markedly only at the first harvest date M1, M2 and M12 ; 2 those that were similar in profile, but genes were more highly expressed in one or the other cultivar M3, M4, and M9 ; and 3 those that had somewhat dissimilar overall profiles with expression levels being greater in Kanlow leaves than in Summer leaves M7, M8 and M Kanlow orange lines and Summer blue lines over the time course of the experiment.

In each panel, relative expression based on module eigengenes is on the Y-axis and harvest times on the X-axis. Total numbers of genes in each module are indicated in parenthesis. Predicted proteins encoded by genes that were part of each co-expression module were queried for their association with the Kyoto Encyclopedia of Genes and Genomes KEGG pathways [ 60 ].

Enrichment of KEGG pathways was variable across each module, with some modules having greater numbers of significantly enriched pathways S1 Data. Within M1, 18 KEGG pathways were significantly enriched and this enrichment appears to be driven by the developmental status of the Kanlow 4 th leaf at the time of the first harvest.

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Assume i holds, i. It suffices to prove M is test-rigid; see [45, 1. Then it follows from 2. Now assume ii holds, i. So M is a strong test module 4. Araya, Iima and Takahashi; see [1, 1. In view of Corollary 4. References [1] T. Araya, K. Iima, and R. On the left perpendicular category of the modules of finite projective dimension. Araya and Y. Remarks on a depth formula, a grade inequality and a conjecture of Auslander. Algebra, 26 11 —, Modules over unramified regular local rings. Illinois J. Auslander and M. Stable module theory. Memoirs of the American Mathematical Society, No. American Mathematical Society, Providence, R.

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